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A-Level Pure Math Exam Paper

The document is the front cover and first 3 questions of a mathematics exam. Question 1 asks to sketch a graph and solve an inequality. Question 2 asks to find the gradient of a parametric curve at a point. Question 3 asks to find values of a and b in a polynomial given divisibility and remainder conditions.

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سحر خالق

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0% found this document useful (0 votes)

111 views20 pages

A-Level Pure Math Exam Paper

Uploaded by

سحر خالق

We take content rights seriously. If you suspect this is your content, claim it here.

Available Formats

Download as PDF, TXT or read online on Scribd

You are on page 1/ 20

Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*9146949640*

MATHEMATICS 9709/32
Paper 3 Pure Mathematics 3 October/November 2023

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INSTRUCTIONS
³ Answer all questions.
³ Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
³ Write your name, centre number and candidate number in the boxes at the top of the page.
³ Write your answer to each question in the space provided.
³ Do not use an erasable pen or correction fluid.
³ Do not write on any bar codes.
³ If additional space is needed, you should use the lined page at the end of this booklet; the question
number or numbers must be clearly shown.
³ You should use a calculator where appropriate.
³ You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
³ Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.

INFORMATION
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 20 pages. Any blank pages are indicated.

JC23 11_9709_32/2R
© UCLES 2023 [Turn over
2

BLANK PAGE

© UCLES 2023 9709/32/O/N/23

1 (a) Sketch the graph of y = 4x − 2. [1]

(b) Solve the inequality 1 + 3x < 4x − 2. [4]

........................................................................................................................................................

© UCLES 2023 9709/32/O/N/23 [Turn over

2 The parametric equations of a curve are

x = ln t2 , y = e2−t ,
2

for t > 0.

Find the gradient of the curve at the point where t = e, simplifying your answer. [4]

................................................................................................................................................................

© UCLES 2023 9709/32/O/N/23

3 The polynomial 2x3 + ax2 − 11x + b is denoted by p x. It is given that p x is divisible by 2x − 1
and that when p x is divided by x + 1 the remainder is 12.

Find the values of a and b. [5]

................................................................................................................................................................

© UCLES 2023 9709/32/O/N/23 [Turn over

4 (a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying the inequalities z − 4 − 3i ≤ 2 and Re z ≤ 3. [4]

(b) Find the greatest value of arg z for points in this region. [2]

........................................................................................................................................................

© UCLES 2023 9709/32/O/N/23

x x + 1
6
Find the exact value of Ô
x2 + 4
5 dx. [6]
0

................................................................................................................................................................

© UCLES 2023 9709/32/O/N/23 [Turn over

6 (a) By sketching a suitable pair of graphs, show that the equation

cot x = 2 − cos x
has one root in the interval 0 < x ≤ 12 π. [2]

(b) Show by calculation that this root lies between 0.6 and 0.8. [2]

........................................................................................................................................................

© UCLES 2023 9709/32/O/N/23

9
Q P
(c) Use the iterative formula xn+1 = tan −1
1
2 − cos xn
to determine the root correct to 2 decimal

places. Give the result of each iteration to 4 decimal places. [3]

........................................................................................................................................................

© UCLES 2023 9709/32/O/N/23 [Turn over

7 (a) By expressing 31 as 21 + 1, prove the identity cos 31 4 cos3 1 − 3 cos 1. [3]

........................................................................................................................................................

© UCLES 2023 9709/32/O/N/23

(b) Hence solve the equation

cos 31 + cos 1 cos 21 = cos2 1

for 0Å ≤ 1 ≤ 180Å. [5]

........................................................................................................................................................

© UCLES 2023 9709/32/O/N/23 [Turn over

2 + 3ai
= , 2 − i, where a and , are real constants.
a + 2i
8 It is given that

(a) Show that 3a2 + 4a − 4 = 0. [4]

........................................................................................................................................................

© UCLES 2023 9709/32/O/N/23

(b) Hence find the possible values of a and the corresponding values of ,. [3]

........................................................................................................................................................

© UCLES 2023 9709/32/O/N/23 [Turn over

9
y

M
x
O a π

The diagram shows the curve y = sin x cos 2x, for 0 ≤ x ≤ π, and a maximum point M , where x = a.
The shaded region between the curve and the x-axis is denoted by R.

(a) Find the value of a correct to 2 decimal places. [5]

........................................................................................................................................................

(b) Find the exact area of the region R, giving your answer in simplified form. [4]

........................................................................................................................................................

10 The equations of the lines l and m are given by

−2
` a ` a ` a ` a
3 1 6
l: r = −2 + , 1 and m: r = −3 + - 4 ,
1 2 6 c
where c is a positive constant. It is given that the angle between l and m is 60Å.

(a) Find the value of c. [4]

........................................................................................................................................................

(b) Show that the length of the perpendicular from 6, −3, 6 to l is 11.

[5]

........................................................................................................................................................

11 The variables x and y satisfy the differential equation

+ y2 + y = 0.
dy
x2
dx
It is given that x = 1 when y = 1.

(a) Solve the differential equation to obtain an expression for y in terms of x. [8]

........................................................................................................................................................

(b) State what happens to the value of y when x tends to infinity. Give your answer in an exact form.
[1]

........................................................................................................................................................

Additional Page

If you use the following lined page to complete the answer(s) to any question(s), the question number(s)
must be clearly shown.

........................................................................................................................................................................

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.

To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge
Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download
at www.cambridgeinternational.org after the live examination series.

Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge
Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.

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A-Level Pure Math Exam Paper

Uploaded by

A-Level Pure Math Exam Paper

Uploaded by

Cambridge International AS & A Level

You must answer on the question paper.

You will need: List of formulae (MF19)

This document has 20 pages. Any blank pages are indicated.

© UCLES 2023 9709/32/O/N/23

1 (a) Sketch the graph of y = 4x − 2. [1]

(b) Solve the inequality 1 + 3x < 4x − 2. [4]

© UCLES 2023 9709/32/O/N/23 [Turn over

2 The parametric equations of a curve are

© UCLES 2023 9709/32/O/N/23

Find the values of a and b. [5]

© UCLES 2023 9709/32/O/N/23 [Turn over

© UCLES 2023 9709/32/O/N/23

© UCLES 2023 9709/32/O/N/23 [Turn over

6 (a) By sketching a suitable pair of graphs, show that the equation

© UCLES 2023 9709/32/O/N/23

places. Give the result of each iteration to 4 decimal places. [3]

© UCLES 2023 9709/32/O/N/23 [Turn over

7 (a) By expressing 31 as 21 + 1, prove the identity cos 31  4 cos3 1 − 3 cos 1. [3]

© UCLES 2023 9709/32/O/N/23

(b) Hence solve the equation

cos 31 + cos 1 cos 21 = cos2 1

© UCLES 2023 9709/32/O/N/23 [Turn over

(a) Show that 3a2 + 4a − 4 = 0. [4]

© UCLES 2023 9709/32/O/N/23

© UCLES 2023 9709/32/O/N/23 [Turn over

(a) Find the value of a correct to 2 decimal places. [5]

© UCLES 2023 9709/32/O/N/23

© UCLES 2023 9709/32/O/N/23 [Turn over

10 The equations of the lines l and m are given by

(a) Find the value of c. [4]

© UCLES 2023 9709/32/O/N/23

© UCLES 2023 9709/32/O/N/23 [Turn over

11 The variables x and y satisfy the differential equation

© UCLES 2023 9709/32/O/N/23

© UCLES 2023 9709/32/O/N/23

© UCLES 2023 9709/32/O/N/23

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1 (a) Sketch the graph of y = 4x − 2. [1]

(b) Solve the inequality 1 + 3x < 4x − 2. [4]

7 (a) By expressing 31 as 21 + 1, prove the identity cos 31 4 cos3 1 − 3 cos 1. [3]